×

Smoothed nonparametric tests and approximations of \(p\)-values. (English) Zbl 1407.62153

Authors’ abstract: We propose new smoothed sign and Wilcoxon’s signed rank tests that are based on kernel estimators of the underlying distribution function of the data. We discuss the approximations of the \(p\)-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon’s tests in the sense of Pitman’s asymptotic relative efficiency, and the differences between the ordinary and new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Although the smoothed tests are not distribution-free, making use of the specific kernel enables us to obtain the Edgeworth expansions, being free of the underlying distribution.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Azzalini, A, A note on the estimation of a distribution function and quantiles by a kernel method, Biometrika, 68, 326-328, (1981) · doi:10.1093/biomet/68.1.326
[2] Bickel, P., Götze, F., Van Zwet, W. (1986). The edgeworth expansion for u-statistics of degree two. The Annals of Statistics, 14(4), 1463-1484. · Zbl 0614.62015
[3] Brown, B., Hall, P., Young, G. (2001). The smoothed median and the bootstrap. Biometrika, 88(2), 519-534. · Zbl 0984.62021
[4] Epanechnikov, VA, Non-parametric estimation of a multivariate probability density, Theory of Probability and Its Applications, 14, 153-158, (1969) · doi:10.1137/1114019
[5] García-Soidán, P. H., González-Manteiga, W., Prada-Sánchez, J. (1997). Edgeworth expansions for nonparametric distribution estimation with applications. Journal of Statistical Planning and Inference, 65(2), 213-231. · Zbl 0915.62022
[6] Hájek, J., Šidák, Z., Sen, P. K. (1999). Theory of rank tests. San Diego: Academic press. · Zbl 0944.62045
[7] Huang, Z., Maesono, Y. (2014). Edgeworth expansion for kernel estimators of a distribution function. Bulletin of Informatics and Cybernetics, 46, 1-10. · Zbl 1400.62033
[8] Lee, A. (1990). U-statistics. Theory and practice. New York: Marcel Dekker. · Zbl 0771.62001
[9] Lehmann, E. L., D’abrera, H. (2006). Nonparametrics: Statistical methods based on ranks. New York: Springer. · Zbl 1217.62061
[10] Maesono, Y., Moriyama, T., Lu, M. (2016). Smoothed nonparametric tests and their properties. arXiv preprintarXiv:1610.02145 · Zbl 1407.62153
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.